Trying to see things from Elizabeth's perspective:
I think she had major abandonment issues and lasting mental issues due to the way she grew up. I think that when Logan talked her into having a baby together, perhaps he promised her that they would be this wonderful family, and she thought that she would be able to be happy and move on from a hurtful past where she wasn't loved or wanted. But she wasn't prepared for what motherhood would bring. Not only was she not capable of putting Gabriel's needs first, but she didn't feel much attachment to Gabe, didn't know how to meet his needs and be a good mother. I think this also upset her as it made her feel abnormal and inadequate. I think she also had some hard feelings because Logan DID bond with Gabe and was able to meet his needs and Gabe responded well to Logan where he didn't with her. I think her relationship with Logan suffered because she was no longer Logan's main focus and I think she felt resentful of both Logan and Gabriel because of that. I think she felt that this was supposed to be a fairy tale, and instead, she felt like the odd man out. I think that is the reason for her rages, cutting up Gabe's things etc. She was angry that she was no longer the most important person in Logan's life, and that even as Gabriel's mother, she was not the most important person in HIS life. I think she felt used, like she was nothing more than the means for Logan to get the baby he wanted. I think she was resentful that she didn't get much time with Logan, and when she did, Gabriel was the focus, not her. I think that her rages were only in part due to mental issues, but I also think they were voluntary to a degree. I think she felt that is the only way she was getting Logan's attention and able to show him how unhappy she was. When Logan finally told her he couldn't do this anymore, she lost it. No way could he talk her into having a baby she didn't want, then make the baby more important than her, and finally leave her, and now she has a baby she needs to take care of that she never wanted. I also would wager a guess that she was worried that previous police involvement would eventually lead to Logan having full custody and she would end up paying child support and be tied to Logan for life. I think she felt so desperate at this point that she would do ANYTHING to hurt Logan, to make him pay for ABANDONING her, and to sidetrack his plans of continuing their family without her. The question is: Is ANYTHING killing Gabriel or selling/giving him to strangers?
This is all JMO.
ETA: My support is 100% for Logan, I'm simply trying to see things from what might be EJ's perspective.